By now, no obvious evidence for WIMPs has been observed in both DM direct detections and collider searches. In particular, the recent constraints from LUX [1]}, PandaX-II [2]},
and XENON1T [3]} tell us that the DM interaction cross section with protons and neutrons is extraordinarily tiny, less than about \(10^{-45}-10^{-46}~\rm {cm^{2}}\) , low enough to rule out most of WIMP DM models.
Refs. [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} show that there is a novel mechanism for suppressing the direct detection cross section
if the global \(U(1)\)  symmetry of Higgs portal model is softly broken to \(\mathbb {Z}_2\)  symmetry by a mass term.
Such a suppression is due to a cancellation in the DM-nucleon scattering amplitude between the SM-like and new Higgs boson contributions at zero momentum transfer. Although the cancellation is spoiled at loop level, the scattering contribution from one loop is trivial and the conclusion remains practically unchanged [13]}, [14]}.
